Ploidacot/Omega-pentacot: Difference between revisions

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Created page with "{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=4|Cots=5|Pergen=[P8, P4/5]|Forms=12, 13, 25, 37|Title=Omega-pentacot|Wedgie=5}} '''Omega-pentacot''' is a temperament archetype where the generator is a semitone, five of which stack to form a perfect fourth of 4/3, and the period is a 2/1 octave. Omega-pentacot temperaments usually generate the 1L 11s and 12L 1s MOS structures. Regular temperaments of omega-pentacot are Cluster MOS|cluster tem..."
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=4|Cots=5|Pergen=[P8, P4/5]|Forms=12, 13, 25, 37|Title=Omega-pentacot|Wedgie=5}}
{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=4|Cots=5|Pergen=[P8, P4/5]|Forms=12, 13, 25, 37|Title=Omega-pentacot (delta-pentacot)|Wedgie=5}}
'''Omega-pentacot''' is a temperament archetype where the generator is a semitone, five of which stack to form a perfect fourth of [[4/3]], and the period is a [[2/1]] octave. Omega-pentacot temperaments usually generate the [[1L 11s]] and [[12L 1s]] MOS structures. Regular temperaments of omega-pentacot are [[Cluster MOS|cluster temperaments]] with 12 clusters of notes in an octave.
'''Omega-pentacot''' is a temperament archetype where the generator is a semitone, five of which stack to form a perfect fourth of [[4/3]], and the period is a [[2/1]] octave. Omega-pentacot temperaments usually generate the [[1L 11s]] and [[12L 1s]] MOS structures. Regular temperaments of omega-pentacot are [[Cluster MOS|cluster temperaments]] with 12 clusters of notes in an octave.
== Intervals and notation ==
While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for 1/5 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]), so vvvC# and ^^Db are enharmonic.
{| class="wikitable"
|+ style="font-size: 105%;" | Omega-pentacot intervals (assuming pure fifth and octave)
|-
! #
! Cents
! Notation
! Name
|-
| −30
| 611.730
| F#
| augmented fourth
|-
| −29
| 711.339
| ^^G
|
|-
| −28
| 810.948
| vG#
|
|-
| −27
| 910.557
| ^A
|
|-
| −26
| 1010.166
| vvA#
|
|-
| −25
| 1109.775
| B
| major seventh
|-
| −24
| 9.384
| ^^C
|
|-
| −23
| 108.993
| vC#
|
|-
| −22
| 208.602
| ^D
|
|-
| −21
| 308.211
| vvD#
|
|-
| −20
| 407.820
| E
| major third
|-
| −19
| 507.429
| ^^F
|
|-
| −18
| 607.038
| vF#
|
|-
| −17
| 706.647
| ^G
|
|-
| −16
| 806.256
| vvG#
|
|-
| −15
| 905.865
| A
| major sixth
|-
| −14
| 1005.474
| ^^Bb
|
|-
| −13
| 1105.083
| vB
|
|-
| −12
| 4.692
| ^C
|
|-
| −11
| 104.301
| vvC#
|
|-
| −10
| 203.910
| D
| major second
|-
| −9
| 303.519
| ^^Eb
|
|-
| −8
| 403.128
| vE
|
|-
| −7
| 502.737
| ^F
|
|-
| −6
| 602.346
| vvF#
|
|-
| −5
| 701.955
| G
| perfect fifth
|-
| −4
| 801.564
| ^^Ab
|
|-
| −3
| 901.173
| vA
|
|-
| −2
| 1000.782
| ^Bb
|
|-
| −1
| 1100.391
| vvB
|
|-
| 0
| 0.000
| C
| perfect unison
|-
| 1
| 99.609
| ^^Db
|
|-
| 2
| 199.218
| vD
|
|-
| 3
| 298.827
| ^Eb
|
|-
| 4
| 398.436
| vvE
|
|-
| 5
| 498.045
| F
| perfect fourth
|-
| 6
| 597.654
| ^^Gb
|
|-
| 7
| 697.263
| vG
|
|-
| 8
| 796.872
| ^Ab
|
|-
| 9
| 896.481
| vvA
|
|-
| 10
| 996.090
| Bb
| minor seventh
|-
| 11
| 1095.699
| ^^Cb
|
|-
| 12
| 1195.308
| vC
|
|-
| 13
| 94.917
| ^Db
|
|-
| 14
| 194.526
| vvD
|
|-
| 15
| 294.135
| Eb
| minor third
|-
| 16
| 393.744
| ^^Fb
|
|-
| 17
| 493.353
| vF
|
|-
| 18
| 592.962
| ^Gb
|
|-
| 19
| 692.571
| vvG
|
|-
| 20
| 792.180
| Ab
| minor sixth
|-
| 21
| 891.789
| ^^Bbb
|
|-
| 22
| 991.398
| vBb
|
|-
| 23
| 1091.007
| ^Cb
|
|-
| 24
| 1190.616
| vvC
|
|-
| 25
| 90.225
| Db
| minor second
|-
| 26
| 189.834
| ^^Ebb
|
|-
| 27
| 289.443
| vEb
|
|-
| 28
| 389.052
| ^Fb
|
|-
| 29
| 488.661
| vvF
|
|-
| 30
| 588.270
| Gb
| diminished fifth
|}
A notable feature of omega-pentacot is the small comma, encountered after 12 steps, which represents one-fifth of a Pythagorean comma (or its equivalence, ''inversed'' diminished second). This makes omega-pentacot scales cluster around [[12edo]].


== Temperament interpretations ==
== Temperament interpretations ==
Line 7: Line 326:


Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is [[18/17]], five of them gives 4/3, so the [[quinticular comma]] (1419857/1417176) is tempered out.  
Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is [[18/17]], five of them gives 4/3, so the [[quinticular comma]] (1419857/1417176) is tempered out.  
=== Quintilischis ===
{{See also| Schismatic family }}
In quintilischis, the generator is 18/17, three of which make [[19/16]], five make 4/3, and 40 make [[10/1|10th harmonic]] in the 2.3.5.17.19 subgroup, so [[4624/4617]], [[6144/6137]], and [[6885/6859]] are tempered out. This temperament is a weak extension of [[schismic]], splitting the fourth in five. In the 2.3.5.7.17.19 subgroup, tempering out [[400/399]] (equating 20/19 and 21/20) leads to [[quintilipyth]] (12 & 253), and tempering out [[476/475]] (equating 19/17 with 28/25) leads to [[quintaschis]] (12 & 289). 


=== Quindromeda ===
=== Quindromeda ===
{{Main| Quindromeda family }}
{{Main| Quindromeda family }}


In [[quindromeda]], the generator is 18/17, three generators make [[19/16]], five make 4/3, and 28 make [[5/1|5th harmonic]] in the 2.3.5.17.19 subgroup, so [[1216/1215]], [[1445/1444]], and [[6144/6137]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 169, 181, 193, 205, 217, 229, and 241 }} edos.
In [[quindromeda]], the generator is 18/17, three of which make 19/16, five make 4/3, and 28 make [[5/1|5th harmonic]] in the 2.3.5.17.19 subgroup, so [[1216/1215]], [[1445/1444]], and [[6144/6137]] are tempered out. Equating 225/224 with 256/255 leads to [[quintakwai]] (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.  
 
Equating 225/224 with 256/255 leads to [[quintakwai]] (12 & 193), which tempers out [[400/399]] (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 & 217), which tempers out [[476/475]] (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.  


=== Quintaleap ===
=== Quintaleap ===
{{Main| Quintaleap family }}
{{Main| Quintaleap family }}


In [[quintaleap]], the generator is 18/17, three generators make 19/16, five make 4/3, and 16 make [[5/2]] in the 2.3.5.17.19 subgroup, so [[256/255]], [[361/360]], and [[4624/4617]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 109, 121, 133, 145, and 157 }} edos.
In [[quintaleap]], the generator is 18/17, three of which make 19/16, five make 4/3, and 16 make [[5/2]] in the 2.3.5.17.19 subgroup, so [[256/255]], [[361/360]], and [[4624/4617]] are tempered out. In the 2.3.5.7.17.19 subgroup, tempering out 400/399 leads to [[quintupole]] (12 & 121), and tempering out 476/475 leads to [[quinticosiennic]] (12 & 145).
 
In the 2.3.5.7.17.19 subgroup, tempering out 400/399 (equating 20/19 and 21/20) leads to [[quintupole]] (12 & 121), and tempering out 476/475 (equating 19/17 with 28/25) leads to [[quinticosiennic]] (12 & 145).


=== Passion ===
=== Passion ===
{{Main| Passion family }}
{{Main| Passion family }}


In [[passion]], the generator is [[16/15]], four generators make [[5/4]], and five make 4/3. It is best tuned with a slightly flat generator of about 98.7{{c}}, and follows that both 3 and 5 should be tuned sharp. The canonical mapping of 7 places [[7/4]] at 10 generators, and follows that the generator should be tuned flatter (about 98.1{{c}}).
In [[passion]], the generator is [[16/15]], four of which make [[5/4]], and five make 4/3. It is best tuned with a slightly flat generator of about 98.7{{c}}, and follows that both 3 and 5 should be tuned sharp. The canonical mapping of 7 places [[7/4]] at 10 generators, and follows that the generator should be tuned flatter (about 98.1{{c}}).
 
=== Ripple ===
{{Main| Ripple family }}
 
In [[ripple]], the generator is [[27/25]], five of which make 4/3, and eight make [[8/5]]. It is best tuned with a sharp generator of about 101–102{{c}}, giving the [[11L 1s]] MOS structure (rather than 1L 11s), and follows that 3 should be tuned flat.


{{Todo| unify precision }}
[[Category:Ploidacots|Omega-pentacot]]

Latest revision as of 07:29, 10 January 2026

< Ploidacot
Omega-pentacot (delta-pentacot)
Pergen [P8, P4/5]
Numeral form 4-sheared 5-cot
Pure generator size 99.61 ¢
Pure period size 1200 ¢
Forms 12, 13, 25, 37
Characteristic multival entry 5

Omega-pentacot is a temperament archetype where the generator is a semitone, five of which stack to form a perfect fourth of 4/3, and the period is a 2/1 octave. Omega-pentacot temperaments usually generate the 1L 11s and 12L 1s MOS structures. Regular temperaments of omega-pentacot are cluster temperaments with 12 clusters of notes in an octave.

Intervals and notation

While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for 1/5 of an inversed diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the Pythagorean comma), so vvvC# and ^^Db are enharmonic.

Omega-pentacot intervals (assuming pure fifth and octave)
# Cents Notation Name
−30 611.730 F# augmented fourth
−29 711.339 ^^G
−28 810.948 vG#
−27 910.557 ^A
−26 1010.166 vvA#
−25 1109.775 B major seventh
−24 9.384 ^^C
−23 108.993 vC#
−22 208.602 ^D
−21 308.211 vvD#
−20 407.820 E major third
−19 507.429 ^^F
−18 607.038 vF#
−17 706.647 ^G
−16 806.256 vvG#
−15 905.865 A major sixth
−14 1005.474 ^^Bb
−13 1105.083 vB
−12 4.692 ^C
−11 104.301 vvC#
−10 203.910 D major second
−9 303.519 ^^Eb
−8 403.128 vE
−7 502.737 ^F
−6 602.346 vvF#
−5 701.955 G perfect fifth
−4 801.564 ^^Ab
−3 901.173 vA
−2 1000.782 ^Bb
−1 1100.391 vvB
0 0.000 C perfect unison
1 99.609 ^^Db
2 199.218 vD
3 298.827 ^Eb
4 398.436 vvE
5 498.045 F perfect fourth
6 597.654 ^^Gb
7 697.263 vG
8 796.872 ^Ab
9 896.481 vvA
10 996.090 Bb minor seventh
11 1095.699 ^^Cb
12 1195.308 vC
13 94.917 ^Db
14 194.526 vvD
15 294.135 Eb minor third
16 393.744 ^^Fb
17 493.353 vF
18 592.962 ^Gb
19 692.571 vvG
20 792.180 Ab minor sixth
21 891.789 ^^Bbb
22 991.398 vBb
23 1091.007 ^Cb
24 1190.616 vvC
25 90.225 Db minor second
26 189.834 ^^Ebb
27 289.443 vEb
28 389.052 ^Fb
29 488.661 vvF
30 588.270 Gb diminished fifth

A notable feature of omega-pentacot is the small comma, encountered after 12 steps, which represents one-fifth of a Pythagorean comma (or its equivalence, inversed diminished second). This makes omega-pentacot scales cluster around 12edo.

Temperament interpretations

Quinticular

See also: Quinticular comma

Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is 18/17, five of them gives 4/3, so the quinticular comma (1419857/1417176) is tempered out.

Quintilischis

See also: Schismatic family

In quintilischis, the generator is 18/17, three of which make 19/16, five make 4/3, and 40 make 10th harmonic in the 2.3.5.17.19 subgroup, so 4624/4617, 6144/6137, and 6885/6859 are tempered out. This temperament is a weak extension of schismic, splitting the fourth in five. In the 2.3.5.7.17.19 subgroup, tempering out 400/399 (equating 20/19 and 21/20) leads to quintilipyth (12 & 253), and tempering out 476/475 (equating 19/17 with 28/25) leads to quintaschis (12 & 289).

Quindromeda

Main article: Quindromeda family

In quindromeda, the generator is 18/17, three of which make 19/16, five make 4/3, and 28 make 5th harmonic in the 2.3.5.17.19 subgroup, so 1216/1215, 1445/1444, and 6144/6137 are tempered out. Equating 225/224 with 256/255 leads to quintakwai (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to quintagar (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.

Quintaleap

Main article: Quintaleap family

In quintaleap, the generator is 18/17, three of which make 19/16, five make 4/3, and 16 make 5/2 in the 2.3.5.17.19 subgroup, so 256/255, 361/360, and 4624/4617 are tempered out. In the 2.3.5.7.17.19 subgroup, tempering out 400/399 leads to quintupole (12 & 121), and tempering out 476/475 leads to quinticosiennic (12 & 145).

Passion

Main article: Passion family

In passion, the generator is 16/15, four of which make 5/4, and five make 4/3. It is best tuned with a slightly flat generator of about 98.7 ¢, and follows that both 3 and 5 should be tuned sharp. The canonical mapping of 7 places 7/4 at 10 generators, and follows that the generator should be tuned flatter (about 98.1 ¢).

Ripple

Main article: Ripple family

In ripple, the generator is 27/25, five of which make 4/3, and eight make 8/5. It is best tuned with a sharp generator of about 101–102 ¢, giving the 11L 1s MOS structure (rather than 1L 11s), and follows that 3 should be tuned flat.

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